The Relative Frequency
Interpretation of Probability
Chapter 7
In situations that we can imagine repeating
many times, we define the probability of a specific
outcome as the proportion of times it would occur
over the long run -- called the relative frequency
of that particular outcome.
Probability
part 1 - review
Found in two ways:
1.  Use a (mathematical) model and calculate probability
(e.g. basic Mendelian genetics)
2.  Estimate from collected data
Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc.
Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc.
Example 7.4 The Probability of Lost Luggage
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The Personal Probability Interpretation
“1 in 176 passengers on U.S. airline carriers
will temporarily lose their luggage.”
Personal probability of an event = the degree
to which a given individual believes the event
will happen.
Ways to express the relative frequency of lost luggage:
•  The proportion of passengers who lose their
luggage is 1/176 or about .006.
•  About 0.6% of passengers lose their luggage.
•  The probability that a randomly selected
passenger will lose his/her luggage is about .006.
•  The probability that you will lose your luggage
is about .006.
Sometimes subjective probability used because the
degree of belief may be different for each individual.
Accuracy? MARGIN OF ERROR:
Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc.
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Conditional Probabilities
•  compliments
P(A) + P(AC) = 1
•  mutually exclusive (disjoint): they do not
contain any of the same outcomes
•  independent: two events are independent if
knowing that one will occur (or has
occurred) does not change the probability
that the other occurs
•  dependent events
Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc.
Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc.
Conditional probability of the event B,
given that the event A occurs,
is the long-run relative frequency with which
event B occurs when circumstances are such
that A also occurs; written as P(B|A).
P(B) = unconditional probability event B occurs.
P(B|A) = “probability of B given A”
= conditional probability event B occurs given
that we know A has occurred or will occur.
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Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc.
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Determining a
Conditional Probability
Rule 4 (conditional probability):
P(B|A) = P(A and B)/P(A)
P(A|B) = P(A and B)/P(B)
Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc.
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Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc.
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Sampling with and without Replacement
Example: Suppose that events A and B are
mutually exclusive with probability
P(A)=1/2 and probability P(B)=1/3.
•  A sample is drawn with replacement if
individuals are returned to the eligible pool
for each selection.
•  A sample is drawn without replacement if
sampled individuals are not eligible for
subsequent selection.
(a) Are A and B independent?
(b) Are A and B complementary events?
Example: Consider randomly selecting two left-handed
students from a class of 30, where 3 students are left-handed.
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Example: When a fair die is tossed, each of the six
sides (numbers 1 through 6) are equally likely to
land face up. Two fair dice, one red and one green,
are tossed.
(a). A = red die is a 3; B = red die is a 6
(b). A = red die is a 3; B = green die is a 6
(c). A = red die and green die sum to 4; B = red
die is a 3.
(d). A = red die and green die sum to 4; B = red
die is a 4.
In which case(s) are A and B disjoint? Independent?
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Example: Suppose you toss a fair coin 6
times. Which of the following sequences is
the most likely?
HTHTTH
HHHTTT
HHHHTH
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Example 7.14: Suppose that there has been a
crime and it is known that that the criminal is a
person within a population of 6,000,000.
Further, suppose that it is known that in this
population only about one person in a million
has a DNA type that matches the DNA found at
the crime scene, so assume that there are six
people in the population whose DNA would
match. An individual in custody has matching
DNA. What is the probability that he is
innocent?
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Los Angeles Times (August 24, 1987):
Several studies of sexual partners of people infected
with [HIV] show that a single act of unprotected
vaginal intercourse has a surprisingly low risk of
infecting the uninfected partner – perhaps one in
100 to one in 1000. For an average, consider the
risk to be one in 500. If there are 100 acts of
intercourse with an infected partner, the odds of
infection increase to one in five.
Statistically, 500 acts of intercourse with one infected
partner or 100 acts of intercourse with five partners
lead to a 100% probability of infection
(statistically, not necessarily in reality).
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